Companion Guide to ‘INSERT TITLE HERE’
0.1 How to use this Companion Guide
Citations
Please use the following citation when using this website: Goin DE, Riddell CA (2022). INSERT TITLE HERE. Pre-print. ADD LINK.
Bug reports
Submit any bug reports to: c.riddell [AT] berkeley [DOT] edu or open an issue on Github.
0.2 Overview
In this companion guide, we illustrate scenarios in which difference in differences (DID) analysis can be applied. Importantly, this is a companion guide – meaning, it does not stand alone. We assume readers know what a DID design is and has some background knowledge outlined in the corresponding manuscript and previous literature. Reviewing the manuscript and its references is a great place to start if this guide leaves you with more questions than answers.
We start with the simplest of difference in differences scenarios and slowly amp up the complexity. For each scenario we describe a target parameter to estimate and the parameter estimated by a two-way fixed effects (TWFE) model. We also apply the Goodman-Bacon decomposition to this parameter to determine if the TWFE estimate is partially composed of estimates that are “forbidden” (e.g., ones that compare a newly treated state to a previously treated state) and apply alternate estimation approaches.
The goal is to illustrate scenarios where the usual TWFE method of estimation provides suitable results and scenarios where this approach is biased. In the latter case, we show alternative methods to estimation to overcome the issue.
In the following examples, the effect of state policy changes on a health outcome is the effect of interest.
0.2.1 Load the R packages used in this analysis
First, load the packages for reading and plotting the data. The
packages bacondecomp,
did, and staggered are specific to DID
analyses.
# you will need to install these packages if you don't have them already
# install.packages(c("here", "readxl", "tidyverse", "patchwork", "magrittr",
# "broom", "ggrepel", "bacondecomp", "staggered", "devtools"))
# devtools::install_github("bcallaway11/did")
# as of July 11, 2022, package `did` is currently unavailable on CRAN, so you
# need to install the developer version from GitHub, using the code above.
# You may also need to download the developer version of `DRDID` before if you
# get an error. To do so use:
# devtools::install_github("pedrohcgs/DRDID")
library(here) #nice file paths
library(readxl) #read in excel data
library(tidyverse) #collection of packages for data science
library(patchwork) #"stiches" together ggplots
library(magrittr) #pipes
library(broom) #tidy displays
library(ggrepel) #for labelling points on a ggplot
library(bacondecomp) #Goodman Bacon Decomposition
library(did) #Callaway and Sant'Anna estimator for DID
library(staggered) #Sun and Abraham estimator for DID0.2.2 Package Versions
Since the packages used for these analyses are relatively new, we anticipate them to change over time and that these changes may affect the readers ability to reproduce the results. Here are the packages we are using in the analyses, if your results differ, it may be because you are using newer versions of the packages.
packageVersion("bacondecomp")## [1] '0.1.1'packageVersion("staggered")## [1] '1.1'packageVersion("did")## [1] '2.2.0.901'0.3 Scenario 1 (2 states by 2 time points)
We first consider the simplest case in which there is one state that never adopts the policy (the never-treated group), and one state that does implement the policy (the treated group). The groups are observed at two time periods only.
Below, the data are read in from an Excel spreadsheet. You can open
the spreadsheet in Excel to view it if that comes naturally to you, or,
use an R View()er window after you have imported it. The
data contain four variables state, time,
policy, and outcome.
state: ID for the grouping variabletime: Time index, begins at 1policy: Indicator variable for exposure to the policy changeoutcome: The outcome variable
s1 <- read_xlsx(here("data", "scenarios.xlsx"), sheet = "scen1",
col_types = c("text", "text", "text", "numeric"))
# str(s1)
# View(s1)Note that I specified the column types for each variable. I read in the state, time, and policy variables as “text” so that R will consider them as categorical/factor variables in the analysis (or, using econ-speak, as “fixed effects”).
0.3.1 Visualization of Scenario 1
The following can be read from the labelled plot:
- Pre-treatment difference of 1
- Post-treatment difference of 3
- Causal effect of policy = 3-1 = 2 or equivalently,
- State 1 difference of 5
- State 2 difference of 3
- Causal effect of policy = 5-3 = 2 under the following assumptions
Three standard assumptions are typical stated in DID papers and required for the DID estimate [in the simple case?] to estimate the causal effect of interest:
- Parallel trends Had the treated group not been treated, the trends for the treated state would continue to be the same as the never-treated state during the post treatment time period.
- No anticipation: The effect of the policy change begins after it is introduced; there is no anticipatory effect before the policy is introduced. Anticipatory effects occur if individuals change their behavior in anticipation of a policy change.
- [NAME FOR THIS ONE:] The policy is the only thing that changes at the time of treatment that affects the outcome.
If you’re familiar with causal inference theory, you will also know about the main assumptions required when estimating causal effects. Ben-Micheal et al. expound on these in their paper, and we expand on them here:
Consistency: The policy change is the same across treated units. For example, if states introduced a texting ban while driving and these bans were associated with different penalties, then the policy change would not meet the consistency assumption because the different penalties may have varyed effects on the outcome.
No interference: Outcomes of individuals in one state do not affect outcomes of individuals in other states.
Exchangeability: Had the treated states been untreated, they would have experienced the trend in outcomes experienced by the untreated group. This causal assumption is the same as the parallel trends assumption. [DANA DO YOU AGREE?]
Positivity: Each state has a non-zero probability of being treated. DISCUSS: See Ben-Micheal paper - they say that positivity is not a required assumption for standard DID]
Correct model specification: SOMETHING ABOUT LINEARITY? ALSO RELATED TO THE PARALLEL TRENDS ASSUMPTION
0.3.2 Two-way fixed effects (TWFE) regression model
Under the canonical TWFE design, we can estimate the policy effect by including a fixed effect (FE) for state, a FE for time, and an indicator for the policy change. The effect estimate is the regression coefficient on the policy variable.
s1_mod <- lm(outcome ~ state + time + policy, data = s1)
tidy(s1_mod)## # A tibble: 4 × 5
## term estimate std.error statistic p.value
## <chr> <dbl> <dbl> <dbl> <dbl>
## 1 (Intercept) 9 NaN NaN NaN
## 2 state2 1.00 NaN NaN NaN
## 3 time2 3 NaN NaN NaN
## 4 policy1 2.00 NaN NaN NaNThe coefficient on the policy term is 2 showing that the regression model estimate equalled the causal effect.
Note: that in this scenario and the other simple
scenarios we cover, there is no variability in the data, so the model
cannot estimate a standard error (as indicated by the NaN
in the regression output). In a later scenario, we add noise to the data
and show how to estimate the standard error.
Takeaway: When there are only two treatment groups and two time points, where one of the groups becomes treated, you can calculate the DID estimate by hand or using TWFE regression.
0.4 Scenario 2A and 2B (3 states, homogeneous and heterogeneous treatment effects)
This scenario is similar to Scenario 1, except now there are two states that undergo treatment. In Scenario 2A, the magnitude of the treatment effect is the same in both states. In Scenario 2B, the magnitude of the treatment effect varies by state.
s2a <- read_xlsx(here("data", "scenarios.xlsx"), sheet = "scen2a",
col_types = c("text", "text", "text", "numeric"))
s2b <- read_xlsx(here("data", "scenarios.xlsx"), sheet = "scen2b",
col_types = c("text", "text", "text", "numeric"))
s2a %<>% mutate(ever_trt = case_when(state %in% c("2", "3") ~ "treated",
state == "1" ~ "never-treated"))
s2b %<>% mutate(ever_trt = case_when(state %in% c("2", "3") ~ "treated",
state == "1" ~ "never-treated"))0.4.1 Visualization of Scenario 2
The following can be read from the labelled plot for Scenario 2A:
- Post-Pre Difference for never-treated state 1: 12-9=3
- Post-Pre Difference for treated state 2: 15-10=5
- Post-Pre Difference for treated state 3: 16-11=5
- Causal effect of policy = 5-3 for both states for an average treatment effect on the treated (ATT) of 2
The following can be read from the labelled plot for Scenario 2B:
- Post-Pre Difference for never-treated state 1: 12-9=3
- Post-Pre Difference for treated state 2: 15-10=5
- Post-Pre Difference for treated state 3: 17-11=6
- Causal effect of policy = 2 for state 2 vs 1 and 3 for state 3 vs 1 for an ATT of 2.5
0.4.2 TWFE Regression model
s2a_mod <- lm(outcome ~ policy + state + time, data = s2a)
tidy(s2a_mod)## Warning in summary.lm(x): essentially perfect fit: summary may be unreliable## # A tibble: 5 × 5
## term estimate std.error statistic p.value
## <chr> <dbl> <dbl> <dbl> <dbl>
## 1 (Intercept) 9 0 Inf 0
## 2 policy1 2.00 0 Inf 0
## 3 state2 1.00 0 Inf 0
## 4 state3 2.00 0 Inf 0
## 5 time2 3 0 Inf 0The coefficient estimate for the policy variable equals to 2, which is the ATT for Scenario 2a.
s2b_mod <- lm(outcome ~ policy + state + time, data = s2b)
tidy(s2b_mod)## # A tibble: 5 × 5
## term estimate std.error statistic p.value
## <chr> <dbl> <dbl> <dbl> <dbl>
## 1 (Intercept) 9 0.500 18.0 0.0353
## 2 policy1 2.5 0.866 2.89 0.212
## 3 state2 0.750 0.661 1.13 0.460
## 4 state3 2.25 0.661 3.40 0.182
## 5 time2 3 0.707 4.24 0.147The coefficient estimate for the policy variable equals to 2.5, which is the ATT for Scenario 2b.
Takeaway: When there are multiple treatment groups and two time points, where >1 of the groups becomes treated, you can calculate the DID estimate by hand or using TWFE regression. If treatment effects are heterogeneous across states, then the estimated effect is the average ATT across the states.
Do you have a sense why the state FEs in Scen 2b are a bit off? The FE for state 2 should be 1 and for state 3 should be 2.
0.5 Scenario 3A and 3B (Two states, multiple time points)
In Scenario 3, the number of time periods is increased to incorporate 5 time points before and after a policy change. One never-treated and one treated group are considered. In Scenario 3A, the effect of treatment is constant once treatment is introduced. In Scenario 3B, the effect of treatment increases with time (i.e., it is dynamic).
Because there are multiple time points, when we read in the data we update the column type for the time variable to be numeric. This will help with plotting the data.
s3a <- read_xlsx(here("data", "scenarios.xlsx"), sheet = "scen3a",
col_types = c("text", "numeric", "text", "numeric", "text"))
s3b <- read_xlsx(here("data", "scenarios.xlsx"), sheet = "scen3b",
col_types = c("text", "numeric", "text", "numeric", "text"))In addition to state, time,
policy, and outcome, the data contains another
variable time_since_policy which equals 0 before treatment
(and always equals 0 for the never-treated), and counts the time since
treatment in the treated state.
0.5.1 Visualization of Scenario 3
The following can be read from the labelled plot for Scenario 3A:
- Pre-treatment difference is 17-15=2
- Post-treatment difference is 34-30=4
- DID = 2
The following can be read from the labelled plot for Scenario 3B:
- Pre-treatment difference is 17-15=2
- Post-treatment difference is 38-30=8
- DID = 6
For Scenario 3B the effect is dynamic, and rather than calculating an ATT that aggregates over post-time, we may want to calculate the effect separately by time since treatment. Here, the effect increases over time, suggesting that the policy change takes time to “kick-in”. In contrast, some policies may be associated with large initial effects that attenuate over time. Knowing if a policy introduction is associated with a changing effect over time is therefore important for understanding the real-world effects of the policy.
Thus, in this scenario we first estimate the average treatment effect using a TWFE model, and then we estimate the dynamic effect by extending the TWFE model to include a categorical indicator for time since the policy’s introduction.
0.5.2 TWFE Regression Model
Estimation of the dynamic effect in Scenario 3A. Different from the
previous scenarios, we need to use factor(time) to ensure
time is modeled using indicator variables (i.e., time fixed effects)
since we imported it as a numeric variable in this scenario to make
plotting easier.
s3a_mod <- lm(outcome ~ policy + state + factor(time),
data = s3a)
tidy(s3a_mod)## Warning in summary.lm(x): essentially perfect fit: summary may be unreliable## # A tibble: 12 × 5
## term estimate std.error statistic p.value
## <chr> <dbl> <dbl> <dbl> <dbl>
## 1 (Intercept) 9.00 4.39e-15 2.05e15 3.62e-120
## 2 policy1 2.00 5.07e-15 3.94e14 1.92e-114
## 3 state2 2.00 3.59e-15 5.57e14 1.20e-115
## 4 factor(time)2 3.00 5.67e-15 5.29e14 1.83e-115
## 5 factor(time)3 6.00 5.67e-15 1.06e15 7.16e-118
## 6 factor(time)4 9 5.67e-15 1.59e15 2.79e-119
## 7 factor(time)5 12 5.67e-15 2.12e15 2.80e-120
## 8 factor(time)6 15 6.21e-15 2.41e15 9.73e-121
## 9 factor(time)7 18 6.21e-15 2.90e15 2.26e-121
## 10 factor(time)8 21 6.21e-15 3.38e15 6.59e-122
## 11 factor(time)9 24 6.21e-15 3.86e15 2.26e-122
## 12 factor(time)10 27 6.21e-15 4.34e15 8.83e-123The coefficient of the policy term is 2 showing that the regression model estimate equaled the causal effect.
Estimation of the dynamic effect in Scenario 3B:
s3b_mod <- lm(outcome ~ policy + state + factor(time),
data = s3b)
tidy(s3b_mod)## # A tibble: 12 × 5
## term estimate std.error statistic p.value
## <chr> <dbl> <dbl> <dbl> <dbl>
## 1 (Intercept) 9.00 1.22 7.35 0.0000801
## 2 policy1 6 1.41 4.24 0.00283
## 3 state2 2.00 1.00 2.00 0.0805
## 4 factor(time)2 3.00 1.58 1.90 0.0943
## 5 factor(time)3 6.00 1.58 3.79 0.00528
## 6 factor(time)4 9.00 1.58 5.69 0.000459
## 7 factor(time)5 12 1.58 7.59 0.0000637
## 8 factor(time)6 13.0 1.73 7.51 0.0000689
## 9 factor(time)7 17 1.73 9.81 0.00000976
## 10 factor(time)8 21 1.73 12.1 0.00000198
## 11 factor(time)9 25.0 1.73 14.4 0.000000519
## 12 factor(time)10 29.0 1.73 16.7 0.000000164The coefficient of the policy term is 6 showing that the regression model estimate equaled the causal effect.
0.5.3 TWFE Regression
Model with time_since_policy specification
Rather than using a binary indicator for the policy change, it can be
coded using the time_since_policy variable. If we include
this as a factor variable in the model, then separate effects will be
estimated for each time since treatment.
For Scenario 3A, using this variable should yield 2 for each time since treatment, since the treatment effect is constant over time.
s3a_mod2 <- lm(outcome ~ time_since_policy + state + factor(time),
data = s3a)
tidy(s3a_mod2)## Warning in summary.lm(x): essentially perfect fit: summary may be unreliable## # A tibble: 16 × 5
## term estimate std.error statistic p.value
## <chr> <dbl> <dbl> <dbl> <dbl>
## 1 (Intercept) 9.00 7.57e-15 1.19e15 3.00e-60
## 2 time_since_policy1 2.00 1.51e-14 1.32e14 1.97e-56
## 3 time_since_policy2 2.00 1.51e-14 1.32e14 1.97e-56
## 4 time_since_policy3 2.00 1.51e-14 1.32e14 1.97e-56
## 5 time_since_policy4 2.00 1.51e-14 1.32e14 1.97e-56
## 6 time_since_policy5 2.00 1.51e-14 1.32e14 1.97e-56
## 7 state2 2.00 6.18e-15 3.24e14 5.47e-58
## 8 factor(time)2 3.00 9.77e-15 3.07e14 6.76e-58
## 9 factor(time)3 6.00 9.77e-15 6.14e14 4.22e-59
## 10 factor(time)4 9 9.77e-15 9.21e14 8.34e-60
## 11 factor(time)5 12 9.77e-15 1.23e15 2.64e-60
## 12 factor(time)6 15.0 1.24e-14 1.21e15 2.77e-60
## 13 factor(time)7 18.0 1.24e-14 1.46e15 1.33e-60
## 14 factor(time)8 21.0 1.24e-14 1.70e15 7.20e-61
## 15 factor(time)9 24 1.24e-14 1.94e15 4.22e-61
## 16 factor(time)10 27.0 1.24e-14 2.18e15 2.64e-61Indeed, this is what we see.
For Scenario 3B, the effect increases by 2 units for every unit of time since treatment is initiated so this should be reflected in the coefficient estimates.
s3b_mod2 <- lm(outcome ~ time_since_policy + state + factor(time),
data = s3b)
tidy(s3b_mod2)## Warning in summary.lm(x): essentially perfect fit: summary may be unreliable## # A tibble: 16 × 5
## term estimate std.error statistic p.value
## <chr> <dbl> <dbl> <dbl> <dbl>
## 1 (Intercept) 9.00 1.05e-14 8.56e14 1.12e-59
## 2 time_since_policy1 2.00 2.10e-14 9.51e13 7.33e-56
## 3 time_since_policy2 4.00 2.10e-14 1.90e14 4.58e-57
## 4 time_since_policy3 6.00 2.10e-14 2.85e14 9.05e-58
## 5 time_since_policy4 8.00 2.10e-14 3.80e14 2.86e-58
## 6 time_since_policy5 10.0 2.10e-14 4.76e14 1.17e-58
## 7 state2 2.00 8.58e-15 2.33e14 2.04e-57
## 8 factor(time)2 3.00 1.36e-14 2.21e14 2.51e-57
## 9 factor(time)3 6.00 1.36e-14 4.42e14 1.57e-58
## 10 factor(time)4 9.00 1.36e-14 6.63e14 3.10e-59
## 11 factor(time)5 12.0 1.36e-14 8.84e14 9.82e-60
## 12 factor(time)6 15.0 1.72e-14 8.74e14 1.03e-59
## 13 factor(time)7 18.0 1.72e-14 1.05e15 4.97e-60
## 14 factor(time)8 21.0 1.72e-14 1.22e15 2.68e-60
## 15 factor(time)9 24 1.72e-14 1.40e15 1.57e-60
## 16 factor(time)10 27.0 1.72e-14 1.57e15 9.81e-61For this scenario, the effect estimate is 2 in the first time after treatment, then 4, and so on.
Note that we could have modeled time_since_policy
linearly, however including time_since_policy as a factor
variable allows the estimation of effects to be non-linear over
time.
Takeaway: When there are two treatment groups and multiple pre- and post- time points, you can calculate the DID estimate using TWFE regression when the effect is constant or dynamic. Using a “time since treatment” indicator will estimate effects separately by time since the policy change, allowing you to see how the effect has changed over time.
0.6 Scenario 4 (4 states, hetergenous treatment effects)
Scenario 4 extends Scenario 3 to the multiple group setting. State 1 is never-treated, while states 2-4 undergo treatment at time=6. The effects are not dynamic in time, but they are heterogeneous with some states having a larger treatment effect.
s4 <- read_xlsx(here("data", "scenarios.xlsx"), sheet = "scen4",
col_types = c("text", "numeric", "text", "numeric", "text", "numeric"))0.6.1 Visualization of Scenario 4
The following can be read from the labelled plot:
Pre-post difference for never-treated state 1: 30-15 = 15
Pre-post difference for treated state 2: 34-17 = 17
Pre-post difference for treated state 3: 35-19 = 16
Pre-post difference for treated state 4: 39-21 = 18
\(DID_{2 vs 1} = 17-15 = 2\)
\(DID_{3 vs 1} = 16-15 = 1\)
\(DID_{4 vs 1} = 18-15 = 3\)
\(ATT = 2\)
0.6.2 TWFE Regression Model
s4_mod <- lm(outcome ~ policy + state + factor(time),
data = s4)
tidy(s4_mod)## # A tibble: 14 × 5
## term estimate std.error statistic p.value
## <chr> <dbl> <dbl> <dbl> <dbl>
## 1 (Intercept) 9.00 0.277 32.4 1.45e-22
## 2 policy1 2.00 0.320 6.24 1.31e- 6
## 3 state2 2.00 0.253 7.90 2.24e- 8
## 4 state3 3.50 0.253 13.8 1.71e-13
## 5 state4 6.50 0.253 25.7 5.34e-20
## 6 factor(time)2 3.00 0.310 9.67 4.20e-10
## 7 factor(time)3 6.00 0.310 19.3 5.83e-17
## 8 factor(time)4 9 0.310 29.0 2.44e-21
## 9 factor(time)5 12 0.310 38.7 1.63e-24
## 10 factor(time)6 15.0 0.392 38.2 2.20e-24
## 11 factor(time)7 18.0 0.392 45.9 2.06e-26
## 12 factor(time)8 21.0 0.392 53.5 3.89e-28
## 13 factor(time)9 24.0 0.392 61.2 1.24e-29
## 14 factor(time)10 27.0 0.392 68.8 5.91e-31The coefficient of the policy term is 2 showing that the regression model estimates the ATT.
One of the issues with TWFE estimation is that contrasts between
earlier-treated and later-treated states contribute to the coefficient
estimate of the policy term. While all states are treated at the same
time in this scenario, we can use the Goodman-Bacon decomposition
function bacon() to partition the effect estimate into
weighted component parts.
Before using the bacon() function, we need to create
numeric-coded versions of some of the variables that are stored as
characters:
s4 %<>% mutate(state_n = as.numeric(as.character(state)),
policy_n = as.numeric(as.character(policy)))The bacon function has an argument called
formula, where you list the outcome as a function of the
policy change, i.e., formula = outcome ~ policy_n. Note
that no fixed effects for state or time are included in the formula.
Instead, the time and state fixedeffects are specified by the
time_varand id_var arguments.
s4_bacon <- bacon(formula = outcome ~ policy_n,
data = s4,
id_var = "state_n",
time_var = "time")## type weight avg_est
## 1 Treated vs Untreated 1 2This small table illustrates that 100% of the weight is put on comparisons between treated and untreated states and the ATT equals 2.
For more detail we print the object:
s4_bacon ## treated untreated estimate weight type
## 2 6 99999 2 1 Treated vs Untreatedtreated=6 says that all treated states start treatment
at time = 6 and are compared to untreated states (with a fictitious
treatment time of 99999).
The results from the Goodman-Bacon Decomposition are reassuring and
tell us we can trust the TWFE estimate. For pedagodgical purposes, we
also show the estimate if we applied the Callaway Sant’Anna estimator
using the att_gt function to estimate group-time ATEs.
Like bacon(), att_gt() requires all numeric
variables. It also requires an argument called gname.
gname expects a variable which encodes for each state the
time of first policy change. For never-treated state 1, this variable
equals 0 and for treated states 2-4 this variable equals 6.
s4_cs <- att_gt(yname = "outcome",
tname = "time",
idname = "state_n",
gname = "time_first_treat",
data = s4,
control_group = "notyettreated",
anticipation = 0)## No pre-treatment periods to testm2_ag <- aggte(s4_cs, type="simple")
summary(m2_ag)##
## Call:
## aggte(MP = s4_cs, type = "simple")
##
## Reference: Callaway, Brantly and Pedro H.C. Sant'Anna. "Difference-in-Differences with Multiple Time Periods." Journal of Econometrics, Vol. 225, No. 2, pp. 200-230, 2021. <https://doi.org/10.1016/j.jeconom.2020.12.001>, <https://arxiv.org/abs/1803.09015>
##
##
## ATT Std. Error [ 95% Conf. Int.]
## 2 0.4942 1.0314 2.9686 *
##
##
## ---
## Signif. codes: `*' confidence band does not cover 0
##
## Control Group: Not Yet Treated, Anticipation Periods: 0
## Estimation Method: Doubly RobustHere, we see the ATT is equal to as estimated by the TWFE model.
Ideally, we would also show you how to run the Sun and Abraham
estimator for this scenario but the function to do so,
staggered_sa(), throws an error when trying to estimate the
standard error. Thus, we will show you how to use this function in the
scenario that incorporates noise into the data.
0.7 Scenario 5 (4 states, staggered treatment timing)
Scenario 5 introduces staggered timing. In this example, the treated states introduce the policy at two different time points. The treatment effects are non-dynamic and there are two never-treated states and two ever-treated states.
s5 <- read_xlsx(here("data", "scenarios.xlsx"), sheet = "scen5",
col_types = c("text", "numeric", "text", "numeric", "text", "numeric"))0.7.1 Visualization of Scenario 5
Visualization of all comparisons made by TWFE regression
Contrast 1:
- Pre-post difference for never-treated state 1 is: 43.5-13.5=30
- Pre-post difference for never-treated state 2 is: 44.5-14.5=30
- Implying the average pre-post difference equals 30 for the never-treated states
- Pre-post difference for treated state 3 is 49.5-16.5 = 33
- \(DID_{3 vs 1,2} = 33-30=3\)
Contrast 2:
- Pre-post difference for never-treated state 1 is: 54-24=30
- Pre-post difference for never-treated state 2 is: 55-25=30
- Implying the average pre-post difference equals 30 for the never-treated states
- Pre-post difference for treated state 4 is 66-33= 33
- \(DID_{4 vs 1,2} = 33-30=3\)
Contrast 3:
- Pre-post difference for later-treated state 4 is: 39-22.5=16.5
- Pre-post difference for earlier-treated state 2 is: 36-16.5=19.5
- \(DID_{3 vs 4} = 19.5-16.5=3\)
Contrast 4:
- Pre-post difference for earlier-treated state 2 is: 60-36=24
- Pre-post difference for later-treated state 4 is: 66-39=27
- \(DID_{4 vs 3} = 27-24=3\)
s5_mod <- lm(outcome ~ policy + state + factor(time), data = s5)
tidy(s5_mod)## Warning in summary.lm(x): essentially perfect fit: summary may be unreliable## # A tibble: 24 × 5
## term estimate std.error statistic p.value
## <chr> <dbl> <dbl> <dbl> <dbl>
## 1 (Intercept) 9.00 1.65e-14 5.46e14 0
## 2 policy1 3.00 1.31e-14 2.30e14 0
## 3 state2 1.00 9.43e-15 1.06e14 0
## 4 state3 3.00 1.41e-14 2.13e14 0
## 5 state4 9.00 1.11e-14 8.10e14 0
## 6 factor(time)2 3.00 2.11e-14 1.42e14 0
## 7 factor(time)3 6.00 2.11e-14 2.85e14 0
## 8 factor(time)4 9.00 2.11e-14 4.27e14 0
## 9 factor(time)5 12 2.13e-14 5.63e14 0
## 10 factor(time)6 15.0 2.13e-14 7.03e14 0
## # … with 14 more rowsThe coefficient of the policy term is 3 showing that the regression model estimates the ATT.
Using the bacon()function, we can see see the four
comparisons being made and the effect estimates for each comparison. The
weight column in the output shows how much weight each estimate
contributes to the ATT. Here, the treated vs untreated comparisons are
the most heavily weighted.
s5 %<>% mutate(state_n = as.numeric(as.character(state)),
policy_n = as.numeric(as.character(policy)))
#bacon decomp says no problems
#only comparing treated vs untreated and the average effect estimate is 2
s5_bacon <- bacon(outcome ~ policy_n,
data = s5,
id_var = "state_n",
time_var = "time")## type weight avg_est
## 1 Earlier vs Later Treated 0.06715 3
## 2 Later vs Earlier Treated 0.15108 3
## 3 Treated vs Untreated 0.78177 3We can also see a list of each comparison and it’s weight:
s5_bacon## treated untreated estimate weight type
## 2 5 99999 3 0.30695444 Treated vs Untreated
## 3 12 99999 3 0.47482014 Treated vs Untreated
## 6 12 5 3 0.15107914 Later vs Earlier Treated
## 8 5 12 3 0.06714628 Earlier vs Later TreatedFor pedagogical purposes, we also show the estimate if we applied the
Callaway Sant’Anna estimator using the att_gt() function to
estimate group-time ATTs.
s5_cs <- att_gt(yname = "outcome",
tname = "time",
idname = "state_n",
gname = "time_first_treat",
data = s5,
control_group = "notyettreated",
anticipation = 0)## No pre-treatment periods to testNote that I suppressed the warnings by using
warning=FALSE in the R markdown chunk header for this
specific chunk. This is because it prints more than 100 lines of the
same warning “## Warning in max(abs(b/bSigma), na.rm = TRUE): no
non-missing arguments to max; ## returning -Inf”. This warning is
because there is no variability in the the data so the model cannot
estimate a standard error. This is not realistic, and in a later example
we will add in variability. I will suppress the warnings in this
document whenever they are too numerous. In practice, you will want to
display warnings.
s5_cs_ag <- aggte(s5_cs, type="simple")
summary(s5_cs_ag)##
## Call:
## aggte(MP = s5_cs, type = "simple")
##
## Reference: Callaway, Brantly and Pedro H.C. Sant'Anna. "Difference-in-Differences with Multiple Time Periods." Journal of Econometrics, Vol. 225, No. 2, pp. 200-230, 2021. <https://doi.org/10.1016/j.jeconom.2020.12.001>, <https://arxiv.org/abs/1803.09015>
##
##
## ATT Std. Error [ 95% Conf. Int.]
## 3 NA NA NA
##
##
## ---
## Signif. codes: `*' confidence band does not cover 0
##
## Control Group: Not Yet Treated, Anticipation Periods: 0
## Estimation Method: Doubly RobustHere, we see the ATT is equal to 3 as estimated by the TWFE model.
Takeaway: When a policy’s introduction is staggered in time and the treatment effect is constant (not dynamic), TWFE can be used to estimate the treatment effect.
0.8 Scenario 6 (4 states, staggered timing, dynamic effects)
Scenario 6 is similar to Scenario 5 except now the treatment effect increases as time goes on, i.e., it is dynamic. The magnitude of the treatment effect is the same for both treated groups.
s6 <- read_xlsx(here("data", "scenarios.xlsx"), sheet = "scen6",
col_types = c("text", "numeric", "text", "numeric", "text", "numeric"))
Visualization of all comparisons made by TWFE regression
Contrast 1:
- Pre-post difference for never-treated state 1 is: 43.5-13.5=30
- Pre-post difference for never-treated state 2 is: 44.5-14.5=30
- Implying the average pre-post difference equals 30 for the never-treated states
- Pre-post difference for treated state 3 is 57-16.5 = 40.5
- \(DID_{3 vs 1,2} = 40.5-30 = 10.5\)
Contrast 2:
- Pre-post difference for never-treated state 1 is: 54-24=30
- Pre-post difference for never-treated state 2 is: 55-25=30
- Implying the average pre-post difference equals 30 for the never-treated states
- Pre-post difference for treated state 4 is 70-33= 37
- \(DID_{4 vs 1,2} = 37-30 = 7\)
Contrast 3:
- Pre-post difference for later-treated state 4 is: 39-22.5=16.5
- Pre-post difference for earlier-treated state 3 is: 39-16.5=22.5
- \(DID_{3 vs 4} = 22.5-16.5 = 6\)
Contrast 4:
- Pre-post difference for earlier-treated state 3 is: 71-39=32
- Pre-post difference for later-treated state 4 is: 70-39=31
- \(DID_{4 vs 3} = 31-32 = -1\)
Contrasts 1 and 2 are okay because they are clean comparisons between never-treated and treated states. Contrasts 3 and 4 are trickier. Contrast 3 compares the earlier treated state to the later treated state. It is okay because parallel trends holds and it only uses the time before the later state is treated to inform the estimation. Contrast 4 is the problem – it uses post-treatment time from state 3 as the “pre” time for the comparison. The issue is that this post time includes the effects of the treatment, and since the effect is dynamic it does not satisfy the parallel trends assumption. However this contrast still contributes to the TWFE regression estimate!
Let’s take a look…
s6_mod <- lm(outcome ~ policy + state + factor(time),
data = s6)
tidy(s6_mod)## # A tibble: 24 × 5
## term estimate std.error statistic p.value
## <chr> <dbl> <dbl> <dbl> <dbl>
## 1 (Intercept) 8.24 1.28 6.44 2.90e- 8
## 2 policy1 6.80 1.01 6.72 1.02e- 8
## 3 state2 1.00 0.731 1.37 1.77e- 1
## 4 state3 5.96 1.09 5.46 1.11e- 6
## 5 state4 9.09 0.861 10.6 6.23e-15
## 6 factor(time)2 3.00 1.63 1.84 7.17e- 2
## 7 factor(time)3 6.00 1.63 3.67 5.41e- 4
## 8 factor(time)4 9 1.63 5.51 9.51e- 7
## 9 factor(time)5 11.1 1.65 6.68 1.16e- 8
## 10 factor(time)6 14.3 1.65 8.65 6.76e-12
## # … with 14 more rowsThe coefficient of the policy term equals 6.8. We can now use the Goodman Bacon decomposition to see how much weight each of the contrasts contributes to the effect estimate:
s6 %<>% mutate(state_n = as.numeric(as.character(state)),
policy_n = as.numeric(as.character(policy)))
s6_bacon <- bacon(outcome ~ policy_n,
data = s6,
id_var = "state_n",
time_var = "time")## type weight avg_est
## 1 Earlier vs Later Treated 0.06715 6.00000
## 2 Later vs Earlier Treated 0.15108 -1.00000
## 3 Treated vs Untreated 0.78177 8.37423It is more helpful to view the four contrasts separately:
s6_bacon## treated untreated estimate weight type
## 2 5 99999 10.5 0.30695444 Treated vs Untreated
## 3 12 99999 7.0 0.47482014 Treated vs Untreated
## 6 12 5 -1.0 0.15107914 Later vs Earlier Treated
## 8 5 12 6.0 0.06714628 Earlier vs Later TreatedSo we can see the estimates of the treatment effect for each contrast are as we calculated them earlier. Note that 15% of the weight of the TWFE estimate is on Contrast 4 – the one making the improper comparison between an earlier treated and a later treated state.
We can also confirm that you get the TWFE estimate by taking the weighted average of the Goodman-Bacon decomposition components:
#see the TWFE estimate
sum(s6_bacon$estimate*s6_bacon$weight)## [1] 6.798561So we can see that the TWFE regression estimate is biased because it incorporates a contrast that we wouldn’t want to make in practice between an earlier treated and a later treated group. To overcome this, we use the Callaway and Sant’Anna estimator.
Like in the earlier examples we start by running the
att_gt() function. Unlike before, we display the output
from this step using summary(). This output shows the
estimated ATT for each combination of treated state and time. The time
is long because it includes estimates for pre-policy time, which helps
with the evaluation of the parallel trends assumption or to see if there
are any lead effects (“anticipation”) of the treatment on the outcome.
You wouldn’t usually report this entire table, but it worth showing here
to see how the estimated effect is dynamic and that decisions need to be
make about how to report dynamics effects including whether or not to
aggregate the effect, and if so, at what level.
s6_cs <- att_gt(yname = "outcome",
tname = "time",
idname = "state_n",
gname = "time_first_treat",
data = s6,
control_group = "notyettreated",
anticipation = 0)## No pre-treatment periods to testsummary(s6_cs)##
## Call:
## att_gt(yname = "outcome", tname = "time", idname = "state_n",
## gname = "time_first_treat", data = s6, control_group = "notyettreated",
## anticipation = 0)
##
## Reference: Callaway, Brantly and Pedro H.C. Sant'Anna. "Difference-in-Differences with Multiple Time Periods." Journal of Econometrics, Vol. 225, No. 2, pp. 200-230, 2021. <https://doi.org/10.1016/j.jeconom.2020.12.001>, <https://arxiv.org/abs/1803.09015>
##
## Group-Time Average Treatment Effects:
## Group Time ATT(g,t) Std. Error [95% Simult. Conf. Band]
## 5 2 0 NA NA NA
## 5 3 0 NA NA NA
## 5 4 0 NA NA NA
## 5 5 3 NA NA NA
## 5 6 4 NA NA NA
## 5 7 5 NA NA NA
## 5 8 6 NA NA NA
## 5 9 7 NA NA NA
## 5 10 8 NA NA NA
## 5 11 9 NA NA NA
## 5 12 10 NA NA NA
## 5 13 11 NA NA NA
## 5 14 12 NA NA NA
## 5 15 13 NA NA NA
## 5 16 14 NA NA NA
## 5 17 15 NA NA NA
## 5 18 16 NA NA NA
## 5 19 17 NA NA NA
## 5 20 18 NA NA NA
## 12 2 0 NA NA NA
## 12 3 0 NA NA NA
## 12 4 0 NA NA NA
## 12 5 0 NA NA NA
## 12 6 0 NA NA NA
## 12 7 0 NA NA NA
## 12 8 0 NA NA NA
## 12 9 0 NA NA NA
## 12 10 0 NA NA NA
## 12 11 0 NA NA NA
## 12 12 3 NA NA NA
## 12 13 4 NA NA NA
## 12 14 5 NA NA NA
## 12 15 6 NA NA NA
## 12 16 7 NA NA NA
## 12 17 8 NA NA NA
## 12 18 9 NA NA NA
## 12 19 10 NA NA NA
## 12 20 11 NA NA NA
## ---
## Signif. codes: `*' confidence band does not cover 0
##
## Control Group: Not Yet Treated, Anticipation Periods: 0
## Estimation Method: Doubly RobustCallaway and Sant’Anna provide many options for aggregating the
group-time effects. The simplest option is specify
type = simple in the attge() function. This
estimate considered only the clean contrasts (Comparisons 1 and 2 in the
figure above) and combines them by weighting by the total time in the
post-treatment period Is there more to say
about how the weighting is done?. For contrast 1, there are 16
time periods post treatment and for contrast 2, there are 9 periods
post-treatment. Thus this estimate is the weighted average: \(10.5\times(16/25) + 7\times(9/25)=\)
9.24
#aggregate the group time average treatment effects
#type = "simple": weighted average of all group-time average treatment effects
#with weights proportional to the group size
s6_cs_ag <- aggte(s6_cs, type="simple")
summary(s6_cs_ag)##
## Call:
## aggte(MP = s6_cs, type = "simple")
##
## Reference: Callaway, Brantly and Pedro H.C. Sant'Anna. "Difference-in-Differences with Multiple Time Periods." Journal of Econometrics, Vol. 225, No. 2, pp. 200-230, 2021. <https://doi.org/10.1016/j.jeconom.2020.12.001>, <https://arxiv.org/abs/1803.09015>
##
##
## ATT Std. Error [ 95% Conf. Int.]
## 9.24 1.1956 6.8967 11.5833 *
##
##
## ---
## Signif. codes: `*' confidence band does not cover 0
##
## Control Group: Not Yet Treated, Anticipation Periods: 0
## Estimation Method: Doubly RobustAlternatively, we can specify type = "group" to get
separate effect estimates according to time of implementation. Note that
group denotes the time of the policy’s introduction for the
treated states.
s6_cs_ag2 <- aggte(s6_cs, type = "group")## Warning in compute.aggte(MP = MP, type = type, balance_e = balance_e, min_e =
## min_e, : Simultaneous critival value is NA. This probably happened because we
## cannot compute t-statistic (std errors are NA). We then report pointwise conf.
## intervals.summary(s6_cs_ag2)##
## Call:
## aggte(MP = s6_cs, type = "group")
##
## Reference: Callaway, Brantly and Pedro H.C. Sant'Anna. "Difference-in-Differences with Multiple Time Periods." Journal of Econometrics, Vol. 225, No. 2, pp. 200-230, 2021. <https://doi.org/10.1016/j.jeconom.2020.12.001>, <https://arxiv.org/abs/1803.09015>
##
##
## Overall summary of ATT's based on group/cohort aggregation:
## ATT Std. Error [ 95% Conf. Int.]
## 8.75 NA NA NA
##
##
## Group Effects:
## Group Estimate Std. Error [95% Pointwise Conf. Band]
## 5 10.5 NA NA NA
## 12 7.0 NA NA NA
## ---
## Signif. codes: `*' confidence band does not cover 0
##
## Control Group: Not Yet Treated, Anticipation Periods: 0
## Estimation Method: Doubly RobustAlternatively, you can estimate the dynamic effect separately for each time since the treatment has been introduced. Remember that this effect estimation is also done for pre-treatment time, so don’t be alarmed by all the rows in this table!
s6_cs_ag3 <- aggte(s6_cs, type="dynamic")## Warning in compute.aggte(MP = MP, type = type, balance_e = balance_e, min_e =
## min_e, : Simultaneous critival value is NA. This probably happened because we
## cannot compute t-statistic (std errors are NA). We then report pointwise conf.
## intervals.summary(s6_cs_ag3)##
## Call:
## aggte(MP = s6_cs, type = "dynamic")
##
## Reference: Callaway, Brantly and Pedro H.C. Sant'Anna. "Difference-in-Differences with Multiple Time Periods." Journal of Econometrics, Vol. 225, No. 2, pp. 200-230, 2021. <https://doi.org/10.1016/j.jeconom.2020.12.001>, <https://arxiv.org/abs/1803.09015>
##
##
## Overall summary of ATT's based on event-study/dynamic aggregation:
## ATT Std. Error [ 95% Conf. Int.]
## 10.5 NA NA NA
##
##
## Dynamic Effects:
## Event time Estimate Std. Error [95% Pointwise Conf. Band]
## -10 0 NA NA NA
## -9 0 NA NA NA
## -8 0 NA NA NA
## -7 0 NA NA NA
## -6 0 NA NA NA
## -5 0 NA NA NA
## -4 0 NA NA NA
## -3 0 NA NA NA
## -2 0 NA NA NA
## -1 0 NA NA NA
## 0 3 NA NA NA
## 1 4 NA NA NA
## 2 5 NA NA NA
## 3 6 NA NA NA
## 4 7 NA NA NA
## 5 8 NA NA NA
## 6 9 NA NA NA
## 7 10 NA NA NA
## 8 11 NA NA NA
## 9 12 NA NA NA
## 10 13 NA NA NA
## 11 14 NA NA NA
## 12 15 NA NA NA
## 13 16 NA NA NA
## 14 17 NA NA NA
## 15 18 NA NA NA
## ---
## Signif. codes: `*' confidence band does not cover 0
##
## Control Group: Not Yet Treated, Anticipation Periods: 0
## Estimation Method: Doubly RobustIn this case, it is very helpful to plot the effect estimates over time:
ggdid(s6_cs_ag3)
Lastly, you can use type = "calendar" to aggregate the
effects over calendar time. Epidemiologically, this doesn’t make a lot
of sense with dynamic effects because it is mixing effect estimates
across difference times since treatment was introduced. In some cases,
only one comparison is contributing to the estimation (e.g., in calendar
time periods where only one group has introduced the treatment), while
at other points, two comparisons contribute. But this “knowledge” is
lost in the presentation. We don’t show the output from using
type = "calendar" but include the code below in case it
makes sense for your setting.
#calendar time specific
#not recommended for this setting...but CS say they prefer it here
#https://bcallaway11.github.io/did/articles/did-basics.html
s6_cs_ag4 <- aggte(s6_cs, type = "calendar")
summary(s6_cs_ag4)
ggdid(s6_cs_ag4)0.8.1 TWFE with
time_since_policy specification
But what if we model TWFE with the time_since_change
indicators?
s6_mod2 <- lm(outcome ~ time_since_policy + state + factor(time),
data = s6)
tidy(s6_mod2) %>% filter(str_detect(term, "time_since")) %>%
mutate(time = as.numeric(gsub("time_since_policy", "", term))) %>%
arrange(time) ## Warning in summary.lm(x): essentially perfect fit: summary may be unreliable## # A tibble: 16 × 6
## term estimate std.error statistic p.value time
## <chr> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 time_since_policy1 3.00 2.72e-14 1.10e14 0 1
## 2 time_since_policy2 4.00 2.72e-14 1.47e14 0 2
## 3 time_since_policy3 5.00 2.77e-14 1.80e14 0 3
## 4 time_since_policy4 6.00 2.77e-14 2.16e14 0 4
## 5 time_since_policy5 7.00 2.77e-14 2.52e14 0 5
## 6 time_since_policy6 8.00 2.77e-14 2.88e14 0 6
## 7 time_since_policy7 9.00 2.77e-14 3.24e14 0 7
## 8 time_since_policy8 10.0 2.87e-14 3.48e14 0 8
## 9 time_since_policy9 11.0 2.87e-14 3.83e14 0 9
## 10 time_since_policy10 12.0 4.00e-14 3.00e14 0 10
## 11 time_since_policy11 13.0 4.00e-14 3.25e14 0 11
## 12 time_since_policy12 14.0 4.00e-14 3.50e14 0 12
## 13 time_since_policy13 15.0 4.00e-14 3.75e14 0 13
## 14 time_since_policy14 16.0 4.00e-14 4.00e14 0 14
## 15 time_since_policy15 17.0 4.03e-14 4.22e14 0 15
## 16 time_since_policy16 18.0 4.03e-14 4.47e14 0 16In this setting, the regression model still works! The estimates from
the policy indicator variables equal the output from the Callaway
Sant’Anna when specified using type = "dynamic"
Takeway: When the treatment effect is staggered and
dynamic, you can still capture the effect estimate using a TWFE model so
long as the policy effect is modeled using the
time since treatment variable. The key question for the
researcher is identifying their parameter of interest – are you
interested in estimating a dynamic effect, or does a parameter that
summarizes over treatment time make sense? I am inclined to start by
estimating a dynamic effect to see if the model supports the presence of
a dynamic effect. If so, it is important of how the policy operates
after being rolled out. If there is no strong support for a dynamic
effect, then I would consider estimating the effects separately by
timing of introduction if that is sensible for the policy change under
study (i.e., if there are only a few separate time points), or
estimating an overall summary parameter.
0.9 Scenario 7 (4 states, staggered timing, dynamic and hetergenous treatment effects)
Scenario 7 is like Scenario 6 except now the treatment effect is dynamic AND heterogeneous. In this scenario, the state that implements the policy change first has a stronger treatment effect (i.e., a steeper change in slope).
s7 <- read_xlsx(here("data", "scenarios.xlsx"), sheet = "scen7",
col_types = c("text", "numeric", "text", "numeric", "text", "numeric"))
Visualization of all comparisons made by TWFE regression
Contrast 1:
- Pre-post difference for never-treated state 1 is: 43.5-13.5=30
- Pre-post difference for never-treated state 2 is: 44.5-14.5=30
- Implying the average pre-post difference equals 30 for the never-treated states
- Pre-post difference for treated state 3 is 64.5-16.5 = 48
- \(DID_{3 vs 1,2} = 48-30 = 18\)
Contrast 2:
- Pre-post difference for never-treated state 1 is: 54-24=30
- Pre-post difference for never-treated state 2 is: 55-25=30
- Implying the average pre-post difference equals 30 for the never-treated states
- Pre-post difference for treated state 4 is 68-33= 35
- \(DID_{4 vs 1,2} = 35-30 = 5\)
Contrast 3:
- Pre-post difference for later-treated state 4 is: 39-22.5=16.5
- Pre-post difference for earlier-treated state 3 is: 42-16.5=25.5
- \(DID_{3 vs 4} = 25.5-16.5 = 9\)
Contrast 4:
- Pre-post difference for earlier-treated state 3 is: 82-42=40
- Pre-post difference for later-treated state 4 is: 68-39=29
- \(DID_{4 vs 3} = 29-40 = -11\)
Similar to the previous scenario, parallel trends is satisfied for contrasts 1-3, so we can be okay with these contrasts contributing to the average effect estimate. Like last time, contrast 4 is the problem, and here the problem is intensified because the “control” state – which was treated in an earlier period – is undergoing a large treatment effect in the “pre” period (times 5-11). This makes the pre-post difference in the control state larger than the pre-post difference in the comparison state – because the control state is still experiencing a stronger (i.e., steeper slope) treatment effect than the state that is treated at time 12. This makes the DID for this group and time look negative – like the treatment is associated with a decrease in the outcome!
Let’s take a look at the regression output:
s7_mod <- lm(outcome ~ policy + state + factor(time),
data = s7)
tidy(s7_mod)## # A tibble: 24 × 5
## term estimate std.error statistic p.value
## <chr> <dbl> <dbl> <dbl> <dbl>
## 1 (Intercept) 6.98 2.76 2.53 0.0143
## 2 policy1 6.84 2.18 3.13 0.00276
## 3 state2 1.00 1.58 0.634 0.529
## 4 state3 11.9 2.35 5.07 0.00000472
## 5 state4 8.17 1.86 4.40 0.0000496
## 6 factor(time)2 3.00 3.53 0.851 0.399
## 7 factor(time)3 6.00 3.53 1.70 0.0944
## 8 factor(time)4 9.00 3.53 2.55 0.0135
## 9 factor(time)5 11.0 3.57 3.09 0.00308
## 10 factor(time)6 14.5 3.57 4.07 0.000147
## # … with 14 more rowsThe coefficient of the policy term equals 6.84.
The question is, does this estimate come close to what it should be? Well, above we calculated effect estimates for four separate contrasts, where the fourth contrast did not satisfy the parallel trends assumption, but the other three contrasts did. So if I were summarizing across these clean contrasts I would want to make a summary estimate across the DID estimates of 18, 5, and 9. One question is, how much weight do we put on each of the estimates? Another question is, do we consider the DID estimate of 9 comparing earlier to later treated states? It isn’t straightforward what the estimate should be when summarizing across several contrasts. Let’s consider the Goodman Bacon decomposition to see how much each comparison contributed to the TWFE estimate:
s7 %<>% mutate(state_n = as.numeric(as.character(state)),
policy_n = as.numeric(as.character(policy)))
s7_bacon <- bacon(outcome ~ policy_n,
data = s7,
id_var = "state_n",
time_var = "time")## type weight avg_est
## 1 Earlier vs Later Treated 0.06715 9.00000
## 2 Later vs Earlier Treated 0.15108 -11.00000
## 3 Treated vs Untreated 0.78177 10.10429s7_bacon## treated untreated estimate weight type
## 2 5 99999 18 0.30695444 Treated vs Untreated
## 3 12 99999 5 0.47482014 Treated vs Untreated
## 6 12 5 -11 0.15107914 Later vs Earlier Treated
## 8 5 12 9 0.06714628 Earlier vs Later TreatedNotice that our fourth contrast (here shown in row 3 of the second table) contributes 15% of the weight to the estimate. We want an estimator that does not use this contrast!
Below we confirm the weighted estimate across the rows equals the TWFE estimate from the regression:
#see the TWFE estimate
sum(s7_bacon$estimate*s7_bacon$weight)## [1] 6.841727The TWFE estimator is influenced by the forbidden fourth contrast that doesn’t meet the parallel trends assumption, so we’d like an estimator that doesn’t suffer from this concern.
Now that we know how to use the Callaway and Sant’Anna code, we can
apply it to this example as well. Let’s start by running the
att_gt() function. This time we won’t display the output
from summary(), but feel free to uncomment that line in the
R markdown document to take a peak:
s7_cs <- att_gt(yname = "outcome",
tname = "time",
idname = "state_n",
gname = "time_first_treat",
data = s7,
control_group = "notyettreated",
anticipation = 0)## No pre-treatment periods to test#summary(s7_cs)Let’s aggregate the ATTs using type = simple. Callaway
and Sant’Anna’s estimate considers only the clean contrasts of 1 and 2
and combines them by weighting by the total time in the post-treatment
period. For contrast 1, there are 16 time periods post treatment and for
contrast 2, there are 9 periods post-treatment. Thus this estimate is
the weighted average: \(18\times(16/25) +
5\times(9/25)=\) 13.32
#aggregate the group time average treatment effects
#type = "simple": weighted average of all group-time average treatment effects
#with weights proportional to the group size
s7_cs_ag <- aggte(s7_cs, type="simple")
summary(s7_cs_ag)##
## Call:
## aggte(MP = s7_cs, type = "simple")
##
## Reference: Callaway, Brantly and Pedro H.C. Sant'Anna. "Difference-in-Differences with Multiple Time Periods." Journal of Econometrics, Vol. 225, No. 2, pp. 200-230, 2021. <https://doi.org/10.1016/j.jeconom.2020.12.001>, <https://arxiv.org/abs/1803.09015>
##
##
## ATT Std. Error [ 95% Conf. Int.]
## 13.32 4.4407 4.6164 22.0236 *
##
##
## ---
## Signif. codes: `*' confidence band does not cover 0
##
## Control Group: Not Yet Treated, Anticipation Periods: 0
## Estimation Method: Doubly Robust[DANA: CAN YOU CONFIRM YOU AGREE WITH WHAT I WROTE ABOVE? specifically how the simple estimate is calculated.]
Alternatively, we can specify type = "group" to get
separate effect estimates according to time of implementation and see
that these match our by-hand calculations:
s7_cs_ag2 <- aggte(s7_cs, type = "group")## Warning in compute.aggte(MP = MP, type = type, balance_e = balance_e, min_e =
## min_e, : Simultaneous critival value is NA. This probably happened because we
## cannot compute t-statistic (std errors are NA). We then report pointwise conf.
## intervals.summary(s7_cs_ag2)##
## Call:
## aggte(MP = s7_cs, type = "group")
##
## Reference: Callaway, Brantly and Pedro H.C. Sant'Anna. "Difference-in-Differences with Multiple Time Periods." Journal of Econometrics, Vol. 225, No. 2, pp. 200-230, 2021. <https://doi.org/10.1016/j.jeconom.2020.12.001>, <https://arxiv.org/abs/1803.09015>
##
##
## Overall summary of ATT's based on group/cohort aggregation:
## ATT Std. Error [ 95% Conf. Int.]
## 11.5 NA NA NA
##
##
## Group Effects:
## Group Estimate Std. Error [95% Pointwise Conf. Band]
## 5 18 NA NA NA
## 12 5 NA NA NA
## ---
## Signif. codes: `*' confidence band does not cover 0
##
## Control Group: Not Yet Treated, Anticipation Periods: 0
## Estimation Method: Doubly RobustAlternatively, you can estimate the dynamic effect separately for each time since the treatment has been introduced. Remember that this effect estimation is also done for pre-treatment time, so don’t be alarmed by all the rows in this table!
s7_cs_ag3 <- aggte(s7_cs, type="dynamic")## Warning in compute.aggte(MP = MP, type = type, balance_e = balance_e, min_e
## = min_e, : Simultaneous conf. band is somehow smaller than pointwise one
## using normal approximation. Since this is unusual, we are reporting pointwise
## confidence intervalssummary(s7_cs_ag3)##
## Call:
## aggte(MP = s7_cs, type = "dynamic")
##
## Reference: Callaway, Brantly and Pedro H.C. Sant'Anna. "Difference-in-Differences with Multiple Time Periods." Journal of Econometrics, Vol. 225, No. 2, pp. 200-230, 2021. <https://doi.org/10.1016/j.jeconom.2020.12.001>, <https://arxiv.org/abs/1803.09015>
##
##
## Overall summary of ATT's based on event-study/dynamic aggregation:
## ATT Std. Error [ 95% Conf. Int.]
## 16.3125 NA NA NA
##
##
## Dynamic Effects:
## Event time Estimate Std. Error [95% Pointwise Conf. Band]
## -10 0.0 NA NA NA
## -9 0.0 NA NA NA
## -8 0.0 NA NA NA
## -7 0.0 NA NA NA
## -6 0.0 NA NA NA
## -5 0.0 NA NA NA
## -4 0.0 NA NA NA
## -3 0.0 NA NA NA
## -2 0.0 NA NA NA
## -1 0.0 NA NA NA
## 0 2.0 0.7413 0.5471 3.4529 *
## 1 3.5 1.1120 1.3206 5.6794 *
## 2 5.0 1.4826 2.0942 7.9058 *
## 3 6.5 1.8533 2.8677 10.1323 *
## 4 8.0 4.4478 -0.7175 16.7175
## 5 9.5 2.5946 4.4148 14.5852 *
## 6 11.0 2.9652 5.1883 16.8117 *
## 7 12.5 3.3359 5.9618 19.0382 *
## 8 14.0 3.7065 6.7354 21.2646 *
## 9 21.0 NA NA NA
## 10 23.0 NA NA NA
## 11 25.0 NA NA NA
## 12 27.0 NA NA NA
## 13 29.0 NA NA NA
## 14 31.0 NA NA NA
## 15 33.0 NA NA NA
## ---
## Signif. codes: `*' confidence band does not cover 0
##
## Control Group: Not Yet Treated, Anticipation Periods: 0
## Estimation Method: Doubly RobustNote that because the treatment effect is heterogeneous (the later treated state has a smaller treatment effect than the earlier treated state) and because there is less observed treated time for the later state, this means that the treatment effect estimates for event times 0 through 8 combine across the two treated states but for times 9 and over those effect estimates are based only on the first treated state. Without knowing this, it looks like the treatment effect “jumps” between event times 8 and 9 and that there is a slope increase, but this is an artifact based on combining across states with heterogeneous effects with different lengths of observed treatment times.
We can also plot the effect estimates over time and see this jump and slope increase between event times 8 and 9. Note also the confidence interval estimates that occurred for times 0 through 8 since there were two data points that could be used to estimate standard errors.
ggdid(s7_cs_ag3)
Here again we don’t display the results using
type = "calendar" but have included the code here in case
you want to run it.
s7_cs_ag4 <- aggte(s7_cs, type = "calendar")
summary(s7_cs_ag4)
ggdid(s7_cs_ag4)0.9.1 TWFE with
time_since_policy specification
But what if we run the TWFE model with the
time_since_change indicators? Here, I run the regression
and filter() out the policy estimates for easy viewing
since the regression table is so large:
s7_mod2 <- lm(outcome ~ time_since_policy + state + factor(time),
data = s7)
tidy(s7_mod2) %>% filter(str_detect(term, "time_since")) %>%
mutate(time = as.numeric(gsub("time_since_policy", "", term))) %>%
arrange(time) ## # A tibble: 16 × 6
## term estimate std.error statistic p.value time
## <chr> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 time_since_policy1 0.621 1.13 0.547 5.87e- 1 1
## 2 time_since_policy2 2.06 1.13 1.81 7.73e- 2 2
## 3 time_since_policy3 4.11 1.16 3.56 9.55e- 4 3
## 4 time_since_policy4 5.64 1.16 4.88 1.63e- 5 4
## 5 time_since_policy5 7.17 1.16 6.21 2.19e- 7 5
## 6 time_since_policy6 8.70 1.16 7.53 2.96e- 9 6
## 7 time_since_policy7 10.2 1.16 8.85 4.59e-11 7
## 8 time_since_policy8 11.7 1.20 9.77 2.89e-12 8
## 9 time_since_policy9 13.3 1.20 11.1 6.07e-14 9
## 10 time_since_policy10 18.7 1.67 11.2 5.00e-14 10
## 11 time_since_policy11 20.8 1.67 12.5 1.48e-15 11
## 12 time_since_policy12 23.0 1.67 13.8 5.41e-17 12
## 13 time_since_policy13 25.2 1.67 15.1 2.41e-18 13
## 14 time_since_policy14 27.4 1.67 16.4 1.29e-19 14
## 15 time_since_policy15 29.5 1.68 17.6 1.04e-20 15
## 16 time_since_policy16 31.7 1.68 18.9 7.43e-22 16The effect estimates on the policy indicators do not equal an average of the ATTs from the two states for each time period. For this example, this specification gives estimates that are systematically lower than those produced by the Callaway Sant’Anna model.
Takeway: When the treatment effect is staggered, dynamic, and hetergeneous, the TWFE regression specification will produced biased results, as will the specification using a categorical variable for time since treatment. In this case, it is best to use another approach like the Callaway Sant’Anna function.
0.10 Scenario 8 (24 states, staggered timing, dynamic and hetergenous treatment effects)
So far, we have considered toy examples where the outcomes didn’t incorporate noise. This was great for getting started, but doesn’t reflect the imprecision in estimates based on real-world data. Furthermore, we couldn’t estimate standard errors because there was not error in the estimation.
This scenario is more realistic. It increases the number of treated and control units, as is common in analyses across several geographic areas like states. We also added variability to the outcome, with different amounts of variability by state to reflect units containing varying sample sizes.
s8_start <- read_xlsx(here("data", "scenarios.xlsx"), sheet = "scen8",
col_types = c("text", "numeric", "text", "numeric", "text", "numeric"))
# we hid some code here from displaying in the html version of this file where
# we build up this scenario and add noise. Please view the Rmd document to see
# these lines of code!
Visualization of all comparisons made by TWFE regression
We start by running the linear model as we’ve done previously:
s8_mod <- lm(outcome ~ policy + factor(state) + factor(time),
data = all_states)
tidy(s8_mod)## # A tibble: 54 × 5
## term estimate std.error statistic p.value
## <chr> <dbl> <dbl> <dbl> <dbl>
## 1 (Intercept) 7.70 1.55 4.98 8.06e- 7
## 2 policy1 10.1 0.806 12.5 2.59e-32
## 3 factor(state)11 0.126 1.46 0.0862 9.31e- 1
## 4 factor(state)12 -0.449 1.46 -0.307 7.59e- 1
## 5 factor(state)13 -1.04 1.46 -0.711 4.77e- 1
## 6 factor(state)14 -0.430 1.46 -0.294 7.69e- 1
## 7 factor(state)15 -0.0183 1.46 -0.0125 9.90e- 1
## 8 factor(state)20 0.713 1.46 0.488 6.26e- 1
## 9 factor(state)21 -1.75 1.46 -1.20 2.32e- 1
## 10 factor(state)22 0.668 1.46 0.457 6.48e- 1
## # … with 44 more rowsUp until now, we haven’t worried about the estimation of the 95%
confidence interval because our previous datasets didn’t contain any
noise. In practice, we want to estimate confidence intervals and to to
that correctly, we need to account for the clustering on data over time
within state. To do that we use the vcovHC function from
the sandwich package.
#install.packages("sandwich")
library(sandwich)
m2_var <- vcovHC(s8_mod)We can then pull out the variance from the policy1 term
from the variance-covariance matrix and use it to compute Wald type
confidence intervals:
est = summary(s8_mod)$coefficients["policy1", "Estimate"]
lb = summary(s8_mod)$coefficients["policy1", "Estimate"] - 1.96*sqrt(m2_var["policy1","policy1"])
ub = summary(s8_mod)$coefficients["policy1", "Estimate"] + 1.96*sqrt(m2_var["policy1","policy1"])The estimated effect is 10.1 with a 95% CI from 8.4 to 11.8.
We can use the Goodman Bacon decomposition to examine the contrasts contributing to the TWFE estimate. I set up this scenario such that the effects should be the same as for scenario 7, though this time the pre-treatment period is longer than in scenario 7.
The DID contrasts for the four comparisons are:
- Treated at time 15 vs. never-treated: 18
- Treated at time 22 vs. never-treated: 5
- Treated at time 22 vs. post-treatment time for states treated at time 5: -11
- Treated at time 5 vs. pre-treatment time for states treated at time 22: 9
We can compare this to the output from the Goodman Bacon decomposition:
all_states %<>% mutate(policy_n = as.numeric(policy))
s8_bacon <- bacon(outcome ~ policy_n,
data = all_states,
id_var = "state",
time_var = "time")## type weight avg_est
## 1 Earlier vs Later Treated 0.09929 9.48462
## 2 Later vs Earlier Treated 0.06383 -9.69867
## 3 Treated vs Untreated 0.83688 11.64355s8_bacon## treated untreated estimate weight type
## 2 15 99999 17.488998 0.45390071 Treated vs Untreated
## 3 22 99999 4.715617 0.38297872 Treated vs Untreated
## 6 22 15 -9.698670 0.06382979 Later vs Earlier Treated
## 8 15 22 9.484617 0.09929078 Earlier vs Later TreatedWe can see that the estimates displayed in the second table are not too far off from the DID contrasts listed above. For this scenario, 84% of the weight is on the treated vs. untreated and about 10% of the weight is on the earlier vs. the later treated, with only 6% of the weight on the forbidden contrast. In the next section we will see how these weights change if pre-treatment time is shortened.
Compare to Callaway and Sant’Anna’s approach:
s8_cs <- att_gt(yname = "outcome",
tname = "time",
idname = "state",
gname = "time_first_treat",
data = all_states,
control_group = "notyettreated",
anticipation = 0)## Warning in att_gt(yname = "outcome", tname = "time", idname = "state", gname
## = "time_first_treat", : Not returning pre-test Wald statistic due to singular
## covariance matrixs8_cs_ag <- aggte(s8_cs, type="simple")
summary(s8_cs_ag)##
## Call:
## aggte(MP = s8_cs, type = "simple")
##
## Reference: Callaway, Brantly and Pedro H.C. Sant'Anna. "Difference-in-Differences with Multiple Time Periods." Journal of Econometrics, Vol. 225, No. 2, pp. 200-230, 2021. <https://doi.org/10.1016/j.jeconom.2020.12.001>, <https://arxiv.org/abs/1803.09015>
##
##
## ATT Std. Error [ 95% Conf. Int.]
## 12.9667 3.1624 6.7684 19.165 *
##
##
## ---
## Signif. codes: `*' confidence band does not cover 0
##
## Control Group: Not Yet Treated, Anticipation Periods: 0
## Estimation Method: Doubly Robustlb = s8_cs_ag$overall.att - 1.96*s8_cs_ag$overall.se
ub = s8_cs_ag$overall.att + 1.96*s8_cs_ag$overall.se
round(c(s8_cs_ag$overall.att, lb, ub), 2)## [1] 12.97 6.77 19.17The CS estimate is 12.97 with 95% CI 6.77 to 19.17.
The simple ATT is a weighted average (based on the amount of time in the post-treatment period) of the estimated effects for the treated groups. To figure out what the group specific estimate are we can ask for the estimates by group:
s8_cs_ag2 <- aggte(s8_cs, type = "group")
summary(s8_cs_ag2)##
## Call:
## aggte(MP = s8_cs, type = "group")
##
## Reference: Callaway, Brantly and Pedro H.C. Sant'Anna. "Difference-in-Differences with Multiple Time Periods." Journal of Econometrics, Vol. 225, No. 2, pp. 200-230, 2021. <https://doi.org/10.1016/j.jeconom.2020.12.001>, <https://arxiv.org/abs/1803.09015>
##
##
## Overall summary of ATT's based on group/cohort aggregation:
## ATT Std. Error [ 95% Conf. Int.]
## 10.6388 2.0382 6.644 14.6336 *
##
##
## Group Effects:
## Group Estimate Std. Error [95% Simult. Conf. Band]
## 15 18.9527 2.8425 13.3371 24.5682 *
## 22 2.3249 2.3235 -2.2653 6.9152
## ---
## Signif. codes: `*' confidence band does not cover 0
##
## Control Group: Not Yet Treated, Anticipation Periods: 0
## Estimation Method: Doubly RobustAnd we can confirm how the simple weighted estimate was calculated:
s8_cs_ag2$att.egt[1]*(16/25) + s8_cs_ag2$att.egt[2]*(9/25)## [1] 12.96669NOTE: (Was expecting 18 and 5 as the group specific estimates – possible we are off just due to random error - need to rerun with a different seed or with less variability and see if that is true)
Finally, we can look at these effect estimated separately for each time since the intervention began and graph them:
s8_cs_ag3 <- aggte(s8_cs, type="dynamic")
summary(s8_cs_ag3)##
## Call:
## aggte(MP = s8_cs, type = "dynamic")
##
## Reference: Callaway, Brantly and Pedro H.C. Sant'Anna. "Difference-in-Differences with Multiple Time Periods." Journal of Econometrics, Vol. 225, No. 2, pp. 200-230, 2021. <https://doi.org/10.1016/j.jeconom.2020.12.001>, <https://arxiv.org/abs/1803.09015>
##
##
## Overall summary of ATT's based on event-study/dynamic aggregation:
## ATT Std. Error [ 95% Conf. Int.]
## 15.9898 2.7295 10.6401 21.3394 *
##
##
## Dynamic Effects:
## Event time Estimate Std. Error [95% Simult. Conf. Band]
## -20 0.0880 2.2047 -5.5745 5.7505
## -19 -2.9862 2.3094 -8.9177 2.9452
## -18 3.0521 2.3766 -3.0519 9.1561
## -17 -0.8759 2.4799 -7.2451 5.4933
## -16 -3.0049 1.9931 -8.1239 2.1141
## -15 -0.9056 1.4719 -4.6859 2.8748
## -14 3.6489 2.7868 -3.5086 10.8065
## -13 -0.6495 1.5822 -4.7130 3.4141
## -12 0.4784 2.3976 -5.6796 6.6363
## -11 -0.4877 2.3074 -6.4140 5.4387
## -10 1.3585 3.1907 -6.8362 9.5533
## -9 -0.6886 1.8567 -5.4572 4.0800
## -8 1.9187 1.9882 -3.1878 7.0252
## -7 0.9691 1.9081 -3.9316 5.8698
## -6 -2.5743 3.7105 -12.1042 6.9556
## -5 0.9476 2.1647 -4.6122 6.5074
## -4 -1.4939 3.0751 -9.3919 6.4041
## -3 -2.3672 1.8374 -7.0862 2.3518
## -2 1.9938 2.5862 -4.6485 8.6361
## -1 1.4807 2.1824 -4.1245 7.0859
## 0 1.4778 3.5334 -7.5972 10.5528
## 1 4.5488 3.2251 -3.7345 12.8320
## 2 5.4845 2.5425 -1.0455 12.0144
## 3 5.6417 3.3956 -3.0796 14.3629
## 4 6.6845 2.2534 0.8969 12.4721 *
## 5 6.2926 2.9516 -1.2881 13.8733
## 6 11.2668 3.4105 2.5074 20.0262 *
## 7 11.9494 3.2717 3.5466 20.3522 *
## 8 14.9849 2.7132 8.0164 21.9533 *
## 9 20.6918 6.6501 3.6120 37.7715 *
## 10 20.7157 2.0250 15.5149 25.9166 *
## 11 25.9967 3.4110 17.2360 34.7574 *
## 12 27.7174 4.1050 17.1744 38.2604 *
## 13 30.3829 3.5458 21.2761 39.4897 *
## 14 31.0607 2.6938 24.1421 37.9792 *
## 15 30.9403 4.9003 18.3545 43.5261 *
## ---
## Signif. codes: `*' confidence band does not cover 0
##
## Control Group: Not Yet Treated, Anticipation Periods: 0
## Estimation Method: Doubly Robustggdid(s8_cs_ag3)
Now that we have confidence intervals, we can assess the parallel trends assumption by investigating if the pre-treatment confidence intervals overlap the null and are randomly distributed around the zero line. This is true in our example, so we wouldn’t be concerned about violating the parallel trends assumption with these data.
We can also calculate the Sun & Abraham estimator. We couldn’t
run this for the other scenarios because the function throws an error
when if cannot estimate the standard errors for those scenarios. To use
the staggered_sa function, we need to add a new variable
where the time_first_treat variable is set to
Inf for all states that were never-treated:
table(all_states$time_first_treat, useNA = "always")##
## 0 15 22 <NA>
## 360 180 180 0all_states$time_first_treat2 <- all_states$time_first_treat
all_states$time_first_treat2[all_states$time_first_treat == 0] <- Inf
#check that the recoding is as planned:
table(all_states$time_first_treat, all_states$time_first_treat2, useNA = "always")##
## 15 22 Inf <NA>
## 0 0 0 360 0
## 15 180 0 0 0
## 22 0 180 0 0
## <NA> 0 0 0 0For this function, we need to specify the data frame in the
df argument, the clustering unit in the i
argument, the time unit in the t argument. Here we use the
newly-defined time_first_treat2 as the grouping variable
g.
s8_sa <- staggered_sa(df = all_states,
i = "state",
t = "time",
g = "time_first_treat2",
y = "outcome",
estimand = "simple")
s8_sa$estimate## [1] 13.05514The estimate of the treatment effect is 13.0551383 and we can calculate its confidence interval using either of these:
#95% CI
round(c(s8_sa$estimate, s8_sa$estimate - 1.96*s8_sa$se, s8_sa$estimate + 1.96*s8_sa$se), 2)## [1] 13.06 8.90 17.21round(c(s8_sa$estimate, s8_sa$estimate - 1.96*s8_sa$se_neyman, s8_sa$estimate + 1.96*s8_sa$se_neyman), 2)## [1] 13.06 8.71 17.40For this example, the Callaway & Sant’Anna and Sun & Abraham estimates are quite close and have overlapping confidence intervals. They’re both higher than the TWFE estimate because they do not use information based on the forbidden contrast.
0.11 Scenario 9 (same as previous but with less pre-exposure time)
In this scenario, we use the same data from Scenario 8, except we limit the time before the first intervention point to 5 time units, rather than the 15 in Scenario 8. If you look back at the illustration of the previous scenario, this change should not effect the estimates of the effects since…finish
The goal is to see how the different estimators are affected by this change.
all_states2 <- all_states %>% filter(time >= 10)s9_mod <- lm(outcome ~ policy + factor(state) + factor(time),
data = all_states2)
tidy(s9_mod)## # A tibble: 45 × 5
## term estimate std.error statistic p.value
## <chr> <dbl> <dbl> <dbl> <dbl>
## 1 (Intercept) 33.9 1.82 18.7 3.35e-58
## 2 policy1 7.62 1.04 7.32 1.11e-12
## 3 factor(state)11 1.04 1.87 0.556 5.79e- 1
## 4 factor(state)12 -0.669 1.87 -0.358 7.21e- 1
## 5 factor(state)13 -1.18 1.87 -0.629 5.29e- 1
## 6 factor(state)14 -0.475 1.87 -0.254 8.00e- 1
## 7 factor(state)15 0.451 1.87 0.241 8.09e- 1
## 8 factor(state)20 0.261 1.87 0.140 8.89e- 1
## 9 factor(state)21 -1.88 1.87 -1.00 3.16e- 1
## 10 factor(state)22 0.397 1.87 0.212 8.32e- 1
## # … with 35 more rowsAgain, we need to use the vcovHC function to estimate
the SE and obtain the 95% confidence interval for the policy
coefficient:
m2_var <- vcovHC(s9_mod)
est2 = summary(s9_mod)$coefficients["policy1", "Estimate"]
lb2 = summary(s9_mod)$coefficients["policy1", "Estimate"] - 1.96*sqrt(m2_var["policy1","policy1"])
ub2 = summary(s9_mod)$coefficients["policy1", "Estimate"] + 1.96*sqrt(m2_var["policy1","policy1"])The estimated effect is 7.6 with a 95% CI from 5.4 to 9.8. Let’s compare thish with the estimate from the previous Scenario:
| Estimate (95% CI) | |
|---|---|
| Scenario 8 | 10.1 (6.8,19.2) |
| Scenario 9 | 7.6 (5.4,9.8) |
The estimate for Scenario 9 is much lower than that for Scenario 8. Let’s see if the weights from the Bacon decomposition can hint at why.
all_states2 %<>% mutate(policy_n = as.numeric(policy))
s9_bacon <- bacon(outcome ~ policy_n,
data = all_states2,
id_var = "state",
time_var = "time")## type weight avg_est
## 1 Earlier vs Later Treated 0.07384 9.78211
## 2 Later vs Earlier Treated 0.13291 -9.69867
## 3 Treated vs Untreated 0.79325 10.32571s9_bacon## treated untreated estimate weight type
## 2 15 99999 17.822890 0.33755274 Treated vs Untreated
## 3 22 99999 4.772245 0.45569620 Treated vs Untreated
## 6 22 15 -9.698670 0.13291139 Later vs Earlier Treated
## 8 15 22 9.782109 0.07383966 Earlier vs Later TreatedAnd again recall the weights from Scenario 8:
s8_bacon## treated untreated estimate weight type
## 2 15 99999 17.488998 0.45390071 Treated vs Untreated
## 3 22 99999 4.715617 0.38297872 Treated vs Untreated
## 6 22 15 -9.698670 0.06382979 Later vs Earlier Treated
## 8 15 22 9.484617 0.09929078 Earlier vs Later TreatedComparing the weights:
## treated untreated estimate Scenario 8 weight Scenario 9 weight
## 1 15 99999 17.82 0.45 0.34
## 2 22 99999 4.77 0.38 0.46
## 3 22 15 -9.70 0.06 0.13
## 4 15 22 9.78 0.10 0.07
## Weight difference type
## 1 -0.12 Treated vs Untreated
## 2 0.07 Treated vs Untreated
## 3 0.07 Later vs Earlier Treated
## 4 -0.03 Earlier vs Later TreatedWe can see that more weight is put on the states treated at time=22 in Scenario 9 vs. 8 and less weight on the states treated at time=15. This is because the TWFE estimator is a weighted combination, where the weights are partially determined by how close the treatment change is to the middle of the panel, with more weight put on the estimates for groups with changes closer to the middle of the panel. For scenario 8, the states with treatment changes at time 15 are exactly in the middle of the panel (which had 30 time units), but for Scenario 9, with a shortened pre-treatment period, less weight is put on this group as they are further away from the middle of the panel and more weight is put on the group that introduced treatment at time point 22 because this becomes closer to the middle.
This weighting is done by the TWFE but does not likely correspond to any weighting that you, the researcher, would choose. Concerning!
Let’s see how the other estimators are effected by this change:
s9_cs <- att_gt(yname = "outcome",
tname = "time",
idname = "state",
gname = "time_first_treat",
data = all_states2,
control_group = "notyettreated",
anticipation = 0)
s9_cs_ag <- aggte(s9_cs, type="simple")
summary(s9_cs_ag)##
## Call:
## aggte(MP = s9_cs, type = "simple")
##
## Reference: Callaway, Brantly and Pedro H.C. Sant'Anna. "Difference-in-Differences with Multiple Time Periods." Journal of Econometrics, Vol. 225, No. 2, pp. 200-230, 2021. <https://doi.org/10.1016/j.jeconom.2020.12.001>, <https://arxiv.org/abs/1803.09015>
##
##
## ATT Std. Error [ 95% Conf. Int.]
## 12.9667 2.9875 7.1112 18.8222 *
##
##
## ---
## Signif. codes: `*' confidence band does not cover 0
##
## Control Group: Not Yet Treated, Anticipation Periods: 0
## Estimation Method: Doubly RobustThe Callaway Sant’Anna estimate of the ATT is the same as for Scenario 9 (with a slightly lower SE).
What about Sun Abraham’s estimate?
s9_sa <- staggered_sa(df = all_states2,
i = "state",
t = "time",
g = "time_first_treat2",
y = "outcome",
estimand = "simple")
s9_sa$estimate## [1] 13.05514round(c(s8_sa$estimate, s8_sa$estimate - 1.96*s8_sa$se, s8_sa$estimate + 1.96*s8_sa$se), 2)## [1] 13.06 8.90 17.21round(c(s8_sa$estimate, s8_sa$estimate - 1.96*s8_sa$se_neyman, s8_sa$estimate + 1.96*s8_sa$se_neyman), 2)## [1] 13.06 8.71 17.40The Sun Abraham estimate also stays exactly the same!
Takeway: The TWFE estimate is affected by the length of the panel. It places more weight on the constrasts which are closer to the middle of the panel. The Callaway Sant’Anna and Sun Abraham estimates do not have this issue.
0.12 To Do
- Explain how the weight is calculated in the Bacon decomp – when would it place a lot of weight on the forbidden contrasts? I have another R file called Bacon weights where I try and implement the weight formulas but I get stuck…
- When will the weight be negative? I still haven’t figured out what scenario this corresponds to. Possibly in a setting where there are no ever-treated states?
- DISCUSSION OF DIFFERENT TARGET PARAMETERS, AND WHY SOME MIGHT BE OF MORE INTEREST THAN OTHERS
- https://causalinf.substack.com/p/illustrating-the-bias-of-twfe-in?utm_source=url maybe try implementing this example
# s8 <- read_xlsx(here("data", "scenarios.xlsx"), sheet = "scen8",
# col_types = c("text", "text", "text", "numeric"))
#
# s8 %<>% mutate(state_n = as.numeric(as.character(state)),
# policy_n = as.numeric(as.character(policy)))
# s8_bacon <- bacon(formula = outcome ~ policy_n,
# data = s8,
# id_var = "state_n",
# time_var = "time")